Journal of Mathematical Sciences

, Volume 244, Issue 1, pp 22–35 | Cite as

Extremal quasiconformality vs bounded rational approximation

  • Samuel L. KrushkalEmail author


We show that, on most of the hyperbolic simply connected domains, the weighty bounded rational approximation in a natural sup norm is possible only for a very sparse set of holomorphic functions (in contrast to the integral approximation). The obstructions are caused by the features of extremal quasiconformality.


Rational approximation holomorphic function quasiconformal maps quasicircles universal Teichmüller space Schwarzian derivative Strebel point Grunsky coefficients 


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  1. 1.
    L. V. Ahlfors and G. Weill, “ A uniqueness theorem for Beltrami equations,” Proc. Amer. Math. Soc., 13, 975–978 (1962).MathSciNetCrossRefGoogle Scholar
  2. 2.
    A. Amboladze and H. Wallin, “ Rational interpolants with prescribed poles, theory and practice,” Complex Var. Theory Appl., 34(4), 399–413 (1997).MathSciNetGoogle Scholar
  3. 3.
    K. Astala, “Self-similar zippers,” in: Holomorphic Functions and Moduli, Vol. I, edited by D. Drasin et al., Springer, New York, 1988, pp. 61–73.CrossRefGoogle Scholar
  4. 4.
    L. Bers, “An approximation theorem,” J. Anal. Math., 14, 1–4 (1965).MathSciNetCrossRefGoogle Scholar
  5. 5.
    L. Bers, “A non-standard integral equation with applications to quasiconformal mappings,” Acta Math., 116, 113–134 (1966).MathSciNetCrossRefGoogle Scholar
  6. 6.
    C. J. Earle, I. Kra, and S. L. Krushkal, “Holomorphic motions and Teichmüller spaces,” Trans. Amer. Math. Soc., 944, 927–948 (1994).zbMATHGoogle Scholar
  7. 7.
    C. J. Earle and Zong Li, “Isometrically embedded polydisks in infinite dimensional Teichmüller spaces,” J. Geom. Anal., 9, 51–71 (1999).MathSciNetCrossRefGoogle Scholar
  8. 8.
    F. P. Gardiner and N. Lakic, Quasiconformal Teichmüller Theory, Amer. Math. Soc., Providence, RI, 2000.zbMATHGoogle Scholar
  9. 9.
    G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Amer. Math. Soc., Providence, RI, 1969.CrossRefGoogle Scholar
  10. 10.
    H. Grunsky, “Koeffizientenbedingungen für schlicht abbildende meromorphe Funktionen,” Math. Z., 45, 29–61 (1939).MathSciNetCrossRefGoogle Scholar
  11. 11.
    A. Gustafsson, “Approximation with rational interpolants in A -∞ (D),” Comput. Methods Func. Theory, DOI Scholar
  12. 12.
    S. L. Krushkal, Quasiconformal Mappings and Riemann Surfaces, Wiley, New York, 1979.zbMATHGoogle Scholar
  13. 13.
    S. L. Krushkal, “Grunsky coefficient inequalities, Carathéodory metric and extremal quasiconformsal mappings,” Comment. Math. Helv., 64, 650–660 (1989).MathSciNetCrossRefGoogle Scholar
  14. 14.
    S. L. Krushkal, “Strengthened Moser’s conjecture, geometry of Grunsky inequalities and Fredholm eigenvalues,” Central Europ. J. Math., 5(3), 551–580 (2007).CrossRefGoogle Scholar
  15. 15.
    S. L. Krushkal, “Rational approximation of holomorphic functions and geometry of Grunsky inequalities,” Contemp. Math., 455, 219–236 (2008).MathSciNetCrossRefGoogle Scholar
  16. 16.
    S. L. Krushkal, “Strengthened Grunsky and Milin inequalities,” Contemp. Math., 667, 159–179 (2016).MathSciNetCrossRefGoogle Scholar
  17. 17.
    R. Kühnau, “Verzerrungssätze und Koeffizientenbedingungen vom Grunskyschen Typ für quasikonforme Abbildungen,” Math. Nachr., 48, 77–105 (1971).MathSciNetCrossRefGoogle Scholar
  18. 18.
    R. Kühnau, “Wann sind die Grunskyschen Koeffizientenbedingungen hinreichend für Q-quasikonforme Fortsetzbarkeit?” Comment. Math. Helv., 61, 290–307 (1986).MathSciNetCrossRefGoogle Scholar
  19. 19.
    N. Lakic, “Strebel points,” in: Lipa’s Legacy, Amer. Math. Soc., Providence, RI, 2001, pp. 417–431.Google Scholar
  20. 20.
    I. M. Milin, Univalent Functions and Orthonormal Systems, Amer. Math. Soc., Providence, RI, 1977.zbMATHGoogle Scholar
  21. 21.
    Z. Nehari, “The Schwarzian derivative and schlicht functions,” Bull. Amer. Math. Soc., 55, 545–551 (1949).MathSciNetCrossRefGoogle Scholar
  22. 22.
    Chr. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.zbMATHGoogle Scholar
  23. 23.
    K. Strebel, “On the existence of extremal Teichmueller mappings,” J. Anal. Math., 30, 464–480 (1976).MathSciNetCrossRefGoogle Scholar
  24. 24.
    W. P. Thurston, “Zippers and univalent functions,” in: The Bieberbach Conjecture: Proceedings of the Symposium on the Occasion of its Proof, edited by A. Baernstein et al., Amer. Math. Soc., Providence, RI, 1986, pp. 185–197.Google Scholar
  25. 25.
    J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, Amer. Math. Soc., Providence, RI, 1965.zbMATHGoogle Scholar
  26. 26.
    L. Zalcman, Analytic Capacity and Rational Approximation, Springer, Berlin, 1968.CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsBar-Ilan UniversityBar-IlanIsrael
  2. 2.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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